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Question
If \[f \left( x \right) = \sqrt{x^2 + 9}\] , write the value of
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Solution
Given:
Now,
\[f(4) = \sqrt{16 + 9} \]
\[ = \sqrt{25} \]
\[ = 5\]
So,
\[\frac{f(x) - f(4)}{x - 4} = \frac{\sqrt{x^2 + 9} - 5}{x - 4}\]
On rationalising the numerator, we get
\[\frac{f(x) - f(4)}{x - 4} = \frac{\sqrt{x^2 + 9} - 5}{x - 4} \times \frac{\sqrt{x^2 + 9} + 5}{\sqrt{x^2 + 9} + 5} \]
\[ = \frac{x^2 + 9 - 25}{(x - 4) \left( \sqrt{x^2 + 9} + 5 \right)} \]
\[ = \frac{x^2 - 16}{(x - 4) \left( \sqrt{x^2 + 9} + 5 \right)}\]
\[ = \frac{(x + 4)}{\sqrt{x^2 + 9} + 5}\]
Taking limit
\[ = \frac{8}{10} \]
\[ = \frac{4}{5}\]
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